This assignment will count the same as a WeBWorK homework and is due 11/8 at the beginning
of class. I expect clear thinking with COMPLETE sentences and proper grammar. Typing is
not required, but nice handwriting is appreciated.
1.
(5 points) Suppose that lim nan = 1. Prove that
n→∞
an diverges.
n=1
Let lim nan = 1. We must show that
n→∞
∞
P
∞
P
an diverges. If lim nan = 1 then there is some N
n→∞
n=1
such that an ≥ 0 for all n > N because otherwise the limit couldn’t be positive. Let bn = n1 so
∞
∞
P
P
lim abnn = lim nan = 1 < ∞. Now we have two series
an and
bn with an , bn > 0 for n
n→∞
n→∞
n=1
n=1
beyond some N and lim nan = 1. Thus we can use the limit comparison test to say that since
n→∞
∞
P
an .
the harmonic series diverges so must
n=1
2.
(5 points) Prove that if
∞
P
an is a convergent series of positive terms then
n=1
∞
P
ln(1 + an )
n=1
converges.
[Hint: consider the function f (x) = x − ln(1 + x)]
Let
∞
P
an be a convergent series of positive terms. We must show
n=1
∞
P
ln(1 + an ) converges.
n=1
Consider f (x) = x − ln(1 + x) with x > 0. Differentiating with respect to x yields f 0 (x) =
1
> 0 for all x > 0. Also note that f (0) = 0 − ln 1 = 0. These two facts show that
1 − 1+x
∞
P
f (x) > 0 and thus x > ln(1 + x) for all x > 0. Now since
an is a series of positive terms
n=1
an > 0 and thus ln(1 + an ) > 0 with an > ln(1 + an ) for all n. Thus the comparison test can be
∞
∞
P
P
used to say that since
an converges so does
ln(1 + an ).
n=1
3.
n=1
(5 points) Prove that if an > 0 and lim (an )1/n = R then
n→∞
∞
P
an converges if R < 1 and
n=1
diverges if R > 1.
Suppose that lim (an )1/n = R with an > 0. If R < 1 there is some number r with R < r < 1
n→∞
1/n
and some integer N such that |an −R| < r −R for all n > N from the definition of convergence.
1/n
1/n
Then we have that 0 < an − R < r − R which is equivalent to 0 < an < r. Taking this
inequality to the nth power gives us 0 < an < rn . Now we have an > 0 and an < rn for all
n > N . Note that rn is the nth term of a convergent geometric series, and we can use the
∞
P
comparison test to say that
an converges. The argument is the similar if R > 1 to show that
n=1
the series diverges with the change that you compare to a divergent geometric series.